New way of quantitative crack closure decomposition and definition of less scattered material-relevant threshold ΔKth

1 New way of quantitative crack closure decomposition and 1 definition of less scattered material-relevant threshold ΔKth 2 Tomáš Vojteka*, Pavel Pokornýa, Radek Kubíčeka, Michal Jambora, Pavel Hutařa 3 a Institute of Physics of Materials, Czech Academy of Sciences, Žižkova 513/22, 616 00 Brno, Czech Republic 4 * Corresponding author: [email protected] 5 6 Abstract 7 The article presents new methodology to obtain crack closure values based on fatigue crack growth 8 rates, which is in contrast to other methods, such as crack closure measurement or numerical modelling. 9 The advantage is that the true load ratio effect in different materials is respected. Moreover, quantitative 10 decomposition to plasticity-induced crack closure, roughness-induced crack closure and oxide-induced 11 crack closure (OICC) components active in the near-threshold regime is possible. It is emphasized that 12 thresholds measured in humid air using the standard load shedding method are affected and non13 conservative. It is suggested that the threshold ΔKth should be measured in dry air in steels and that in this 14 way it is less scattered, conservative and more relevant as a material parameter, since it does not depend 15 on influencing factors of the test. A simple experimental setup to obtain thresholds in dry air is presented. 16 On the other hand, the evaluated OICC component is responsible for large scatter and dependence of 17 threshold on various testing conditions. Crack growth rate data and fracture surface images are presented 18 for five different steel grades and the new methodology is demonstrated on these data. Surprisingly, the 19 influence of OICC in stainless steels is as significant as in corroding steels. Understanding of the 20 mechanisms and improvement of the threshold reliability and conservativeness contributed to the 21 progress of the long-term discussed problem of applicability of the threshold parameter to residual fatigue 22 life estimations. 23 Key words 24 fatigue crack growth threshold, crack closure, load shedding, air humidity, steels 25 1 Introduction 26 1.1 Fatigue crack growth threshold 27 The fatigue crack growth threshold ΔKth, representing the limit of cyclic loading below which the 28 long crack does not propagate, is the crucial material parameter necessary for residual fatigue life (RFL) 29 estimation of cyclically loaded engineering components [1,2]. This is especially important in applications 30 with high demands on long-term safe operation, where the damage-tolerance design is used. Apart from 31 the distinction between the cases of growing and non-growing crack and the definition of damaging and 32 non-damaging cycles of the loading spectrum, it is also essential to know this parameter accurately for 33 reliable prediction of fatigue crack behaviour [3,4]. While most of the material research focuses on faster 34 crack propagation in the Paris regime, there are many serious conceptual questions urgently needed to be 35 answered about the near-threshold regime [2,5,6]. During the last decades, studies concerning the 36 threshold were done but no significant progress in the topic of the scatter of the threshold as a material 37 parameter was made. Since only small differences of the threshold value may result in drastically large 38 changes of the total calculated RFL, the description of fatigue crack growth (FCG) rate in the near39 threshold regime should be improved significantly. 40 2 It should be emphasized that engineering components are usually designed to be durable for a long 1 time, which means that the vast majority of operational loading cycles lies below or slightly above the 2 fatigue limit. This means that the initiated fatigue cracks will be either non-propagating or propagating in 3 the near-threshold regime [1,3]. Therefore, to achieve reasonable accuracies of RFL estimations, the FCG 4 rate data in the near-threshold regime should be determined as precisely as possible. Unfortunately, many 5 material characterisation studies do not take care of this, perhaps due to experimental and theoretical 6 difficulties associated with the threshold determination. One of the issues is that very slow crack growth 7 demands much experimental time. Another issue is that there are many unresolved questions about the 8 influencing factors on threshold and the related mechanisms, which brings confusion into the topic of 9 threshold measurement techniques [7-9]. The results obtained using just one threshold measurement 10 cannot provide sufficient information about the scatter band, in which the obtained value lies. Such 11 procedures may be dangerously non-conservative. Numerous experimental threshold values produced by 12 researches may seem untrustworthy for the community in applications. Additionally, incorrect threshold 13 values may also affect component testing, since the threshold value determines which loading amplitudes 14 can be omitted from the spectrum as non-damaging. More details about the topic of the omission-level 15 loading cycles can be found in [9]. 16 The urgency of the need for an improvement of the situation makes the topic of threshold highly up17 to-date, which is documented by interest in the topic by different research groups [7,10,11] and it is also 18 by conclusions of the special scientific workshop dedicated to the problems of threshold and its 19 applications, which were published in the overview article [2]. From this article (and in agreement with 20 other sources), the following points can be deduced: (i) It was known for a long time that the standard 21 experimental methods for threshold determination often resulted in a too large scatter of the results, which 22 were sometimes non-conservative. (ii) It was known that the variation was related to crack closure 23 mechanisms, such as plasticity-induced crack closure (PICC), roughness-induced crack closure (RICC) 24 and oxide-induced crack closure (OICC), see also [1,12-14]. (iii) No reliable methodology for 25 quantitative consideration of these components was presented, either for experimental studies or for 26 numerical simulations that would be usable for RFL estimations. Therefore, the development of a new 27 approach is required, which is the aim of the present work. 28 The threshold ΔKth and the crack tip shielding effects should not be overlooked in material properties 29 research. It is usually believed that the material fatigue damage mechanisms and the related resistance to 30 fatigue crack propagation are the same as in the case of large crack growth rates or even monotonic 31 loading. It is usually believed that the phenomena related to material strength are the same as those 32 involved in the resistance to fatigue crack propagation and that the only problem is that they occur at a 33 much smaller scale near the crack tip. This is wrong for two reasons. First, the part of the threshold 34 related to the true material resistance to cracking in front of the crack tip (effective threshold) does not 35 depend on classical material properties, such as the yield stress or the ultimate tensile strength. The 36 effective threshold depends only on the Young modulus and the Burgers vector length [15-19]. Therefore, 37 it is easily predictable in most metals and it cannot solve the problems related to the observed scatter of 38 the measured thresholds ΔKth. Second, due to the small crack opening displacements typically occurring 39 in the near-threshold regime, the crack tip shielding effects become dominant within the total ΔKth value. 40 These effects are also responsible for the observed scatter and the strong dependency of threshold on 41 many experimental influencing factors [1,2], while they have nothing to do with material strength. 42 Therefore, in order to improve the situation, the relevant mechanisms and properties should be studied. In 43 particular, the crack tip shielding effects, such as crack branching or fracture surface contact, should be 44 characterized as precisely as possible, which has the potential of improving RFL simulation results 45 significantly. 46 3 1.2 Major problem of current approaches #1 – measurement techniques 1 It may seem that crack closure has already been studied sufficiently long time in the last decades but 2 it turns out that a working reliable quantitative tool is still awaited to be established. The most commonly 3 used approaches to the load ratio effect description, such as the NASGRO method [20,21], consider PICC 4 as the only mechanism of closure. In the near-threshold regime, it is required to take RICC and OICC into 5 account too. Recent studies indicate that OICC represents a significant component of ΔKth in corroding 6 steels and that it depends on such "exotic" factors as air humidity or loading frequency [5,22], which are 7 factors typically differing very much among different testing facilities, as well as in applications, while 8 they are seldom monitored or controlled. Therefore, application of ΔKth is not straightforward and 9 a number of serious challenges should be resolved in order to obtain realistic RFL estimations. 10 One of the problems is that the threshold measurement is essentially an unnatural procedure for the 11 fatigue crack to emerge, involving application of particular loading history designed for the test, which is 12 very different from actual loading in operation. Thus, it is necessary to understand the mechanisms and 13 the related influencing factors to improve the trustworthiness of the measured data and their 14 transferability to components. Furthermore, it should be identified which of the factors are the most 15 relevant and it should be determined what is the range of possible variation of the resulting data in order 16 to assess their conservativeness. 17 Newman [23] pointed out that thresholds measured by the most widely used standard load shedding 18 method according to the ASTM 647 [24] are affected and non-conservative, supporting this conclusion by 19 both experiments and modelling of the PICC effect. Consequently, he suggested using alternative 20 experimental procedures for obtaining conservative thresholds, taking advantage of the cyclic 21 compression precracking technique [25,26]. This technique was developed to avoid the problems with 22 crack closure effects during testing. In particular, the compression precrack load reduction (CPLR) 23 method followed by the compression precrack constant amplitude (CPCA) procedure seem to be 24 a promising alternative. The precrack produced by cyclic compression-compression loading ensures 25 reduction of the PICC overload effect, which in turn also reduces the OICC effect on threshold during 26 load reduction. Another possibility is to apply the procedure of constant Kmax with increasing Kmin to 27 obtain conservative thresholds [2,27]. In this case, the increasing R ratio leads to obtaining values close to 28 the effective threshold. The thresholds for loading with low load ratios need to be obtained by a different 29 strategy. This can be overcome by the use of a multiple Kmax modification reported in [28]. Some argue 30 that high-R closure can also be present when applying these strategies [29]. 31 The present work emphasizes that the effect of OICC causes significant loading history effects and 32 that they should be considered in any analysis concerning in the near-threshold regime in steels. Using 33 unsuitable methods for materials susceptible to the oxide debris effect can lead to dangerously high 34 thresholds, which may result in extremely long predicted RFLs, while in real operation the situation may 35 be different. The present approach has been developed to avoid such errors and to bring clarity into the 36 topic of quantitative influence of individual crack closure components. 37 38 4 1.3 Major problem of current approaches #2 – numerical modelling 1 An appropriate quantitative description of PICC is a necessary condition for studying and evaluating 2 RICC and OICC. It was reported that the up-to-date models of PICC have limited predictability and that 3 their ability to reproduce the load ratio effect reliably in different materials is limited [30]. The most 4 common prediction is that the PICC effect is nearly the same in all materials, independently of their 5 cyclic properties, which cannot be correct theoretically, and which does not agree with experiments 6 either. 7 Typically, the studies rely on the idea that PICC originates from the plastic stretch of material in the 8 crack tip plastic zone during crack propagation. This is only partially true because only the monotonic 9 loading material properties are considered. In reality, the reversed (cyclic) plastic zone generated under 10 fatigue loading is also essential for understanding PICC. During unloading, the reversed plasticity 11 diminishes the positive plastic stretch of the material. The cyclic plastic material properties determine the 12 extent of the reduction of the plastic stretch formed during loading, in other words, it takes back some of 13 the deformation generated in tension. The theory of PICC should be corrected in this sense [31]. It is the 14 difference between the plastic stretch produced during loading and the compressive rebound realized 15 during unloading, what determines the level of PICC. Inevitably, the amount of the rebound in 16 compression depends on material cyclic behaviour. This explains why the models based only on 17 monotonic material properties have limited ability to reproduce the true behaviour. In general, the model 18 should be able to consider one material model for generating the monotonic plastic zone and a different 19 material model to generate the cyclic plastic zone. Although this may be reproduced in some 20 sophisticated finite element models, the results depend on many parameters, which need to be tuned to 21 match the experiment. Besides, only a small number of cycles can be modelled, which makes this 22 approach both non-predictive and ineffective (too much computational power needed for insufficient 23 number of loading cycles). 24 As a consequence, the presented approach suggests that no modelled PICC value should be imposed 25 on the real material behaviour during fitting of the FCG rate data. Instead, the true load ratio effect 26 exhibited by the material should be derived from experimental data obtained at different load ratios. It 27 was shown that the nearly constant predicted PICC value for all materials overestimated the load ratio 28 effect in materials with cyclic softening and that it underestimated the load ratio effect in materials with 29 cyclic hardening or with supplementary brittle microfracture mechanism [30]. For these reasons, the 30 newly developed methodology is required to enable consideration of the real observed material 31 behaviour, instead of relying on numerical predictions. This concerns (quasi-)constant amplitude loading. 32 For variable amplitude loading, numerical modelling becomes primary, taking the newly defined material 33 parameters as inputs. 34 Using the results of the strip-yield model [32,33] has been a successful way to estimate PICC. In the 35 frame of this model, limited influence of the PICC value was possible for adaptation to the material 36 differences. The constraint factor α could be used for variation of the results, since it is the only parameter 37 influencing them to a sufficient extent. In the present approach, this is rather avoided due to the defined 38 physical meaning of the α factor. It should correspond to the true constraint conditions, such as the 39 specimen and crack geometry, the maximum applied stress and the material yield stress. Moreover, as 40 shown in [30], in order to reproduce the large observed differences between materials, the constraint 41 factor α would have to be varied too much up to nonsensical values. Therefore, the preferred way is to 42 find some other parameter in the strip-yield model for variation of the PICC results. In the present work, 43 the crack closure parameter f = Kcl / Kmax, describing the PICC effect, is simply determined from the 44 experimentally observed load ratio effect in the investigated materials. 45 5 1.4 Oxide-induced crack closure and its significance 1 Although the oxide-induced crack closure was studied already long time ago, e.g. [34,35], no 2 practically useable quantitative description was provided. The situation changed few years ago when it 3 was revealed that the thresholds in steels decrease significantly in dry air due to the elimination of oxide 4 debris production on fracture surfaces, which allowed studying of the quantitative significance of the 5 OICC effect [13,22]. The mechanism of OICC is that the oxide layer on fracture surfaces is disrupted by 6 high compression contact with very high number of repetitions, which is the same mechanism as fretting, 7 and where microscale friction is essentially involved during the high pressure contact. The debris of oxide 8 and metallic surface is accumulated in the crack wake, with the layer thicknesses growing to the order of 9 hundreds of nanometres. Due to the specific volume of the ferrous oxide being larger than that of metallic 10 iron, the crack wake is filled in by extra material. The layer thickness is comparable to the crack opening 11 displacement in the near-threshold regime, which means that the effective ΔK can be reduced 12 significantly. The subsequent crack branching reduce the effective ΔK as well. This is why OICC 13 represents such a significant constituent of the applied threshold and, therefore, of the material fatigue 14 strength. The conditions to the emergence of notable OICC are as follows: a very slow crack growth 15 (<10–6 mm/cycle) and the presence of another crack closure mechanism, such as PICC or RICC. At high 16 load ratio where crack closure is missing, the oxide debris is not generated at any crack growth rates. 17 Some estimations of crack closure components were done for the railway axle steel EA4T [13,22]. 18 The present approach abandons the intention of estimation of the RICC component theoretically. Instead, 19 it uses a new way and the OICC effect is studied on more materials to exclude the possibility of the large 20 OICC influence being an anomaly of the EA4T material. Other studies related to OICC were also 21 published, where more details about the specific problems near threshold can be found. For instance, in 22 [8], the difference between thresholds obtained at 10–7 mm/cycle and at 10–8 mm/cycle was pointed out. 23 In the works [9,14] it was emphasized that OICC generated during non-damaging loading cycles cause 24 significant retardation of the subsequent crack growth, which leads to errors in RFL estimations if these 25 cycles are not considered in simulations. It was reported in [5] that using low loading frequency, such as 26 5 Hz, significantly reduces the threshold value due to the absence of OICC. This represents an issue for 27 testing, since such a low frequency would make testing impractically long, while larger frequencies 28 would lead to non-conservative results. As a consequence, the present work offers a solution to this 29 problem by reducing the OICC, while still keeping the test at high frequency. 30 1.5 Purpose and aims 31 As mentioned above, it is important to know the range, in which the threshold ΔKth can vary in both 32 experimental measurements and in applications. Only small changes cause significant differences in RFL, 33 while the thresholds obtained by the standard experimental measurement techniques can differ 34 significantly. 35 Historically, an important step to analyse ΔKth was its division into the intrinsic and the extrinsic 36 resistance to fatigue crack propagation, in other words, the distinguishing between crack tip shielding, 37 mostly realized by crack closure effects, and the effective threshold ΔKth,eff. A further division of the 38 extrinsic part into individual crack closure mechanisms is the key point to be focused on in order to make 39 a progress in this topic. Understanding of the mechanisms and their quantitative significance is essential 40 for the judgement of the influencing factors occurring during testing and during applications. Such a 41 progress is necessary for improvement of the RFL predictive tools, since the relevant parameters should 42 be taken into account in their development. Therefore, it is required to provide an easily applicable 43 methodology to quantify the constituents of ΔKth regarding the mechanisms. In particular, the purpose of 44 this work is to provide the answers to these four questions: 45 6 (1) What is the composition of the ΔKth value from the point of view of mechanisms and which of 1 them is the largest source of the experimental scatter and disagreement? Which quantitative 2 components of ΔKth reflect properties of the material and which reflect parameters of the test? 3 (2) How to get back to the original meaning of ΔKth as a material parameter that could be used in 4 simulations, free from too many problematic influences? 5 (3) How should ΔKth be experimentally measured to be predominantly conservative and safe for 6 residual fatigue life predictions, while not being too much conservative, as is the case of ΔKeff,th? 7 (4) How significant is the effect of oxide debris accumulation in different corroding and non8 corroding steels? 9 To answer these questions, an advanced experimental program was designed for several materials. All 10 materials are used in important applications with the requirement of long-term reliable operation. Because 11 corrosion is a crucial factor for production of oxide particles, steels with different corrosion resistance 12 were selected to reveal the role of the OICC effect in them. 13 14 2 New methodology of crack growth rate data processing to 15 separate crack closure mechanisms and to find the material16 relevant threshold ΔKth 17 A new methodology is presented with the purpose of avoiding the problems related to uncertainties of 18 crack closure determination, especially in the near-threshold regime. Apart from the improvement of the 19 accuracy of RFL estimations, the methodology aims to provide better understanding of mechanisms, 20 identification of relevant material properties and other influencing factors. 21 2.1 Acquisition of crack growth rate data 22 The procedures of crack growth rate data determination according to standards are often used, e.g. 23 ASTM E647 [24] or ISO 12108 [36]. Several da/dN vs. ΔK data sets should be obtained for various load 24 ratios R in order to have sufficient basis for RFL determination. 25 The NASGRO equation [20,21] fits the experimental crack growth rate data with a pre-defined crack 26 closure parameter f, which has two disadvantages. First, only the plasticity-induced crack closure 27 mechanism is considered, which results in inaccurate ideas about crack closure in the near-threshold 28 regime. Second, very similar f values are predicted for all materials, which is inconvenient for studying of 29 different material properties. For example, it was reported that for a 3D-printed 304L steel, the f 30 parameter failed to describe the dependence of crack growth rates on the load ratio [30]. In the present 31 study, it is suggested that the f values are determined directly from the crack growth rates, which 32 characterizes the material accurately. It is computed based on the crack growth rates measured at two 33 different load ratios, typically R = 0.1 and R = 0.8. In the presented analysis, it is sufficient to work with 34 only these two load ratios. For more complex studies, where f should be expressed for any desired R, the 35 use of the f values predicted by NASGRO is currently the only practical and efficient way to do it. The 36 discrepancies between the real material behaviour and the predicted f can be overcome by determining the 37 f using the strip-yield model simulations with an additional variable, as suggested in [30]. 38 7 In the first step of the described methodology, the following three da/dN vs. ΔK data measurements 1 were done: (i) at R = 0.1 in air with normal relative humidity at room temperature, (ii) at R = 0.1 in air 2 with relative humidity less than 10% at room temperature (absolute humidity less than 2 g/m3) and (iii) at 3 R = 0.8, where air humidity does not matter, however, it is easier to conduct the test in normal (humid) 4 air. The data including the threshold were obtained in this work using the standard ASTM load shedding 5 method, see Section 3.2 for details. After that, the data were fitted using a special procedure described in 6 the next section. 7 2.2 Evaluation of the plasticity-induced crack closure effect 8 2.2.1 Modified fitting procedure 9 The present approach is based on the idea that it is not purposeful to search for the f parameter by 10 means of numerical simulations or compliance change measurement techniques. Reliability of the results 11 of these approaches is questionable [37,38], there are too many influences and the predicted values need 12 to be verified experimentally anyway. Here, instead, the true experimentally observed load ratio effect is 13 extracted from the crack growth rate data and the f parameter is evaluated accordingly. It expresses the 14 load ratio effect directly as the desired value, regardless of the mechanisms involved or their complexity. 15 This method is applied to the Paris regime data to avoid additional crack closure mechanisms that are 16 active near threshold. 17 The NASGRO Equation (1) was used in order to ensure compatibility of the approach with the most 18 commonly used methodologies for RFL estimation. 19 d𝑎 d𝑁 =𝐶[(1−𝑓 1−𝑅)∆𝐾]𝑛(1−∆𝐾th ∆𝐾 )𝑝 (1) 20 Here, a is the crack length, N is the number of applied loading cycles, C, n, p and ΔKth are fitting 21 parameters, f = Kcl/Kmax is the crack closure parameter (note that only PICC is included in f), R = Kmin/Kmax 22 is the load ratio and ΔK is the applied stress intensity factor range. The term associated with the final 23 fracture in the original NASGRO equation is omitted in this work, since it corresponds to insignificant 24 number of loading cycles and since the loading is usually associated with large-scale yielding, making the 25 K parameter invalid. 26 In the newly proposed procedure, the first step is that the three described experimentally obtained 27 FCG rate curves are fitted separately using a simplified form of the equation, where Δ𝐾eff is replaced by 28 ∆𝐾. In this way, the term containing f and R is removed, which allows obtaining of two different C 29 coefficients, one for the high load ratio, typically R = 0.8, which is denoted as Ceff, and one for the low 30 load ratio, R = 0.1 in this work, which is denoted CR=0.1. 31 (𝑑𝑎 𝑑𝑁)𝑅=0.8 =𝐶eff(∆𝐾)𝑛(1−∆𝐾th ∆𝐾 )𝑝 (2) 32 (𝑑𝑎 𝑑𝑁)𝑅=0.1 =𝐶R=0.1(∆𝐾)𝑛(1−∆𝐾th ∆𝐾 )𝑝 (3) 33 The coefficient CR=0.1 is kept equal for the humid air data and for the dry air data. The threshold values are 34 fitted as different for all three curves, while the exponents n and p are kept equal for all three curves. 35 8 2.2.2 Finding of f 1 The shift of the data in the Paris regime due to the load ratio effect can be captured by the values of 2 Ceff and CR=0.1. In Eq. (2), ∆𝐾 corresponds to Δ𝐾eff, hence the Ceff value has the same role as the 3 undistinguished C value in the original NASGRO equation. The presence of PICC at R = 0.1 that can be 4 described by the f value is supposed to shift the data in such a way that they are described by Eq. (3). 5 Therefore, Eq. (3) can also be written in the second form that enables the coefficient CR=0.1 to be 6 mathematically expressed as follows: 7 (𝑑𝑎 𝑑𝑁)𝑅=0.1 =𝐶R=0.1(∆𝐾)𝑛(1−∆𝐾th ∆𝐾 )𝑝=𝐶eff [∆𝐾(1−𝑓 1−𝑅)]𝑛(1−∆𝐾th ∆𝐾 )𝑝 (4) 8 𝐶R=0.1 =𝐶eff (1−𝑓 1−𝑅)𝑛 (5) 9 Then, the searched f value can be derived as: 10 𝑓 =1−(1−𝑅)(𝐶R=0.1 𝐶eff )1 𝑛 (6) 11 Finding of this value enables the further described complete quantitative decomposition of ∆𝐾 into 12 individual crack closure mechanisms. 13 2.2.3 Clarification of quantitative PICC expression 14 In order to correctly describe the PICC effect quantitatively, a special separation has to be first 15 introduced. The part of PICC lying inside of the range of the applied ΔK should be defined. The 16 component of PICC lying below Kmin should be subtracted, since anything lying outside of ΔK = Kmax – 17 Kmin is non-existent and thus irrelevant. Therefore, the total value of PICC corresponding to the parameter 18 f, which is KPICC = f·Kmax, should be divided into two parts. The first part is equal to the PICC effect 19 denoted here as ΔKPICC, lying inside of the loading range ΔK. The second part is simply equal to Kmin or in 20 the case of negative load ratios R it is equal to 0. This is one option which is adopted in the present work 21 for the sake of clarity and not to call the negative part of the loading cycle as PICC. Technically, the 22 second part could always be equal to Kmin including the negative load ratios and only the first line of 23 Eq. (7) could be used. This would simplify the computation of other closure components for negative R, 24 since ΔK would not have to be transformed into Kmax to obtain the correct summing, see also Section 25 4.3.4. 26 The part of PICC lying inside ΔK is then: 27 Δ𝐾PICC ={𝑓𝐾max −𝐾min ……… for 0≤𝑅 <1 𝑓𝐾max ……… for 𝑅 <0 . (7) 28 Considering Kmax = ΔK (1 – R), the expression takes this form: 29 Δ𝐾PICC = Δ𝐾(𝑓−𝑅+ 1−𝑅 ) , (8) 30 where R+ = R for 0 ≤ R < 1 and R+ = 0 for R < 0. Such determined f value is then supposed to be valid for 31 the whole crack growth rate curve. 32 33 9 2.3 Evaluation of the roughness-induced crack closure component 1 The RICC component is the remaining closure part, which cannot be evaluated by any other way than 2 by subtraction of all other components from ΔK. Although it is called RICC, in reality it is the component 3 simply corresponding to the rest of the difference between ΔK and ΔKeff, after PICC and OICC are 4 subtracted, regardless of the true physical mechanism. Based on the data measured in dry air, the 5 decomposition of ΔK is as follows: 6 Δ𝐾 =Δ𝐾eff +Δ𝐾PICC +Δ𝐾RICC (9) 7 and using the expression (8) for ΔKPICC, the term ΔKRICC can be derived as: 8 Δ𝐾RICC =Δ𝐾−Δ𝐾PICC −Δ𝐾eff (10) 9 Δ𝐾RICC =Δ𝐾[1−(𝑓−𝑅+ 1−𝑅 )]−Δ𝐾eff . (11) 10 This value can be used for evaluation of the RICC effect for any desired da/dN and at threshold. 11 In order to graphically express the ΔK component separation, an extra line was defined in the crack 12 growth rate diagrams. It was plotted as the dashed blue line in the schematic Fig. 1 and in the results in 13 Section 3.3. For every crack growth rate da/dN the quantities ΔK and ΔKeff obtained from the 14 experimental fits are used to produce a new value on the horizontal axis. The relationship between this 15 new value and ΔK and ΔKeff is derived from Eq. (9), where ΔKPICC and ΔKeff are grouped together on one 16 side, see Eq. (12), keeping in mind that the sums are for dry air, hence no OICC component is included 17 yet. 18 Δ𝐾eff +Δ𝐾PICC =Δ𝐾−Δ𝐾RICC (12) 19 Both sides of Eq. (12) define the borderline between the area for PICC and the area for RICC in the 20 diagram in Fig. 1. Thus, the graphical separation of all ΔK components can be seen. The left-hand side is 21 more straightforward to express, using Δ𝐾PICC according to Eq. (8). Note that ΔKPICC is not equal to 22 𝑓𝐾max. Thus, for every da/dN in the measured range, the blue dashed line is created by the points 23 satisfying the horizontal axis values defined by Eq. (13). 24 Δ𝐾eff +Δ𝐾(𝑓−𝑅+ 1−𝑅 ) (horizontal axis values of the blue dashed line) (13) 25 26 16 3.2 Measurement of fatigue crack growth rates 1 In this work, the crack closure effects are evaluated form the experimentally obtained fatigue crack 2 growth rate curves. The curve measured at R = 0.8 is presumed to provide ΔKeff data. The curves for 3 R = 0.1 measured in dry and humid air serve to reveal the basic effect of load ratio and oxide debris. The 4 fatigue crack growth rates da/dN were measured using the middle-crack tension M(T) specimens with the 5 width 2W = 60 mm and the thickness B = 5 mm for the steels X20Cr13, GX4CrNi13-4, EA1N and EA4T. 6 In the case of the steel P265GH, the compact tension C(T) specimens with the parameter W = 30 mm and 7 the thickness B = 6 mm were used. Geometry of the specimens is schematically depicted in Fig. 3. 8 A sharp notch was produced by electric discharge machining with the wire diameter of 0.25 mm. The 9 experiments with the M(T) specimens were performed using a resonant testing machine Schenck PVQ 10 with the force capacity of 60 kN. The experiments with the C(T) specimens were performed using a linear 11 motor testing machine Instron electropulse 10000 with the force capacity of 10 kN. All experiments were 12 tested at frequencies in the range from 40 to 80 Hz. 13 The temperature and humidity was controlled in the laboratory. The temperature was set to 23 °C and 14 the absolute humidity was set to 10 grams of water molecules per cubic meter, which corresponds 15 approximately to 50% of relative humidity at 23 °C. The specimens tested in humid air were exposed 16 directly to the controlled laboratory air. The specimens tested in dry air were surrounded by a special 17 chamber for reduction of humidity. The amount of water vapour inside the chamber was reduced by the 18 presence of silica gel particles. The absolute humidity was kept below 2 grams of water molecules per 19 cubic meter, which corresponds to relative humidity lower than 10% at 23 °C. The chambers were 20 originally designed by authors, one for the M(T) specimens, see Fig. 4(a), and one for the C(T) 21 specimens, see Fig. 4(b). The chamber for the M(T) specimens had no mechanical effect on the specimen, 22 which was tested and reported in [13], where more details about the usage of this chamber can also be 23 found. Due to much smaller stiffness and loading force, the chamber for the C(T) specimens had to be 24 mechanically isolated, see the scheme in Fig. 4(b). Relative humidity inside the chamber was monitored 25 using two sensors of the type HIH4000-003 (accuracy of ± 3.5%) and the temperature inside the chamber 26 was measured using the temperature sensor SMT 160-30. 27 28 29 17 1 Fig. 3. Geometry of the M(T) specimen (left) and C(T) specimen (right) used for the 2 experiments. 3 4 5 Fig. 4. Sealed chamber surrounding the specimen during silica experiments in dry air, (a) the 6 M(T) specimen, (b) the C(T) specimen with dynamic isolation. Silica gel particles were 7 located inside the chamber to pump out air moisture. 8 9 10 18 The precracks in the M(T) specimens were initiated at the notch root using the force amplitude 1 Fa = 23 kN at the load ratio R = –1, which resulted in the initial maximum SIFs in the range from 10 to 2 14 MPa·m0.5 depending on the notch length. Note that the whole cycle range at R = –1 contributes to 3 crack initiation in the notch. Therefore, relatively small maximum SIF could be used, which is good to 4 reduce the effect of overload and not to use too large loading force of the testing machine. On the other 5 hand, the C(T) specimens can be loaded only at a positive load ratio. This is why the maximum initial 6 SIFs were in the range from 15 to 16 MPa·m0.5. The precracks were initiated in the notch root using cyclic 7 force with the amplitude Fa = 1.72 kN and the load ratio R = 0.1. 8 During the test, the crack increments were measured optically using digital cameras fixed to 9 a travelling table equipped with the micro-position measurement system with the accuracy of 0.01 mm. 10 The crack length was measured on both specimen surfaces and the values were averaged (two crack 11 lengths for the C(T) specimen, four crack lengths for the M(T) specimen). This methodology has been 12 successfully used for a long time and provided reliable data for many experiments. The advantage of this 13 method is that the crack length is measured directly, and it is not calculated from other variables, such as 14 changes in the electrical resistivity or the mechanical stiffness. 15 After the crack initiated and reached 1 mm, the experiment started with a step-wise load shedding 16 (ΔK-decreasing) procedure to determine the crack growth threshold. The load amplitude reduction 17 corresponded to the parameter C = –0.4 mm–1 defined in the standard ASTM E647 [24]. After the crack 18 stopped at threshold, the force amplitude was increased and the crack was re-propagated with a gradually 19 increasing ΔK until the final fracture. The da/dN-ΔK data were processed according to the standard 20 ASTM E647 in order to exclude invalid data points (e.g. a too large difference between the lengths 21 measured on the two sides). 22 23 3.3 Processed crack growth rate curves and fracture surfaces 24 For each material a set of three crack growth rate curves were measured and presented in Figs. 5 – 9: 25 (i) at R = 0.8 in humid air – green points and solid fitting line, (ii) at R = 0.1 in dry air – blue points and 26 solid fitting line and (iii) at R = 0.1 in humid air – red points and solid fitting line. In addition, a blue 27 dashed line is plotted as a border between PICC and RICC, as explained in Section 2.3. The experimental 28 points at da/dN = 10–9 mm/cycle were plotted for cases when no more crack growth was detected. 29 Based on observation of the fracture surfaces, several statements can be made. At R = 0.1 in humid 30 air, an increasingly significant oxide debris layer was formed and visible optically at the fracture surfaces 31 when approaching the crack arrest at threshold. At R = 0.1 in dry air, the oxide layer was significantly 32 eliminated. A full elimination of the oxide would require the air to be fully dry, the water vapour 33 completely pumped out, which was not possible with the simple technology of silica gel particles. Even 34 with the used technology, the threshold was noticeably smaller in dry air. This was the case of all 35 investigated steels, including the stainless steels, which is quite surprising and remarkable. It reveals that 36 the effect of OICC is not limited to corroding steels. Surface oxidation plays a significant role in 37 corrosion-resistant steels, perhaps due to the repeated mechanical disruption of the passivation layer. 38 The observed clean fracture surfaces at R = 0.8 in humid air confirmed that the enhanced oxide layer 39 was not formed and that crack closure causing repeated contact pressure is the necessary condition for the 40 formation of oxide debris. 41 42 19 1 2 Fig. 5. (a) Experimental crack growth rates and their fitting lines for the X20Cr13 steel obtained 3 at R = 0.8 in humid air (green diamonds), at R = 0.1 in dry air (blue circles) and at R = 0.1 4 in humid air (red crosses) accompanied by the blue dashed line separating the PICC and 5 RICC components of ΔK, as defined in Section 3.3. (b) – (d) light microscopy images of 6 the fracture surfaces of the corresponding specimens. The investigated area of crack 7 propagation is marked by the white vertical arrow and the crack arrest at the measured 8 threshold is underlined by the white dotted line. 9 10 11 1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1 2 4 8 16 32 da/dN[mm/cycle] ΔK[MPa·m0.5] R=0.8 humid air R=0.1 humid air R=0.1 dry air R=0.1 ΔKeff + PICC R = 0.1 dry air R = 0.1 humid air R = 0.8 humid air 1 mm (d) 1 mm (c) 1 mm (b) X20Cr13 R = 0.8 humid air R = 0.1 humid air (a) R = 0.1 dry air R = 0.1 ΔKeff + ΔKPICC 20 1 2 Fig. 6. (a) Experimental crack growth rates and their fitting lines for the GX4CrNi13-4 steel 3 obtained at R = 0.8 in humid air (green diamonds), at R = 0.1 in dry air (blue circles) and 4 at R = 0.1 in humid air (red crosses) accompanied by the blue dashed line separating the 5 PICC and RICC components of ΔK, as defined in Section 3.3. (b) – (d) light microscopy 6 images of the fracture surfaces of the corresponding specimens. The investigated area of 7 crack propagation is marked by the white vertical arrow and the crack arrest at the 8 measured threshold is underlined by the white dotted line. 9 10 11 1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1 2 4 8 16 32 da/dN[mm/cycle] ΔK[MPa·m0.5] R=0.8 humid air R=0.1 humid air R=0.1 dry air R=0.1 ΔKeff + PICC R = 0.1 dry air R = 0.1 humid air R = 0.8 humid air 1 mm (d) 1 mm (c) 1 mm (b) GX4CrNi13-4 R = 0.8 humid air R = 0.1 humid air (a) R = 0.1 dry air R = 0.1 ΔKeff + ΔKPICC 21 1 2 Fig. 7. (a) Experimental crack growth rates and their fitting lines for the EA1N steel obtained at 3 R = 0.8 in humid air (green diamonds), at R = 0.1 in dry air (blue circles) and at R = 0.1 in 4 humid air (red crosses) accompanied by the blue dashed line separating the PICC and 5 RICC components of ΔK, as defined in Section 3.3. (b) – (d) light microscopy images of 6 the fracture surfaces of the corresponding specimens. The investigated area of crack 7 propagation is marked by the white vertical arrow and the crack arrest at the measured 8 threshold is underlined by the white dotted line. 9 10 11 1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1 2 4 8 16 32 da/dN[mm/cycle] ΔK[MPa·m0.5] R=0.8 humid air R=0.1 humid air R=0.1 dry air R=0.1 ΔKeff + PICC R = 0.1 dry air R = 0.1 humid air R = 0.8 humid air 1 mm (d) 1 mm (c) 1 mm (b) EA1N R = 0.8 humid air R = 0.1 humid air (a) R = 0.1 dry air R = 0.1 ΔKeff + ΔKPICC 22 1 2 Fig. 8. (a) Experimental crack growth rates and their fitting lines for the EA4T steel obtained at 3 R = 0.8 in humid air (green diamonds), at R = 0.1 in dry air (blue circles) and at R = 0.1 in 4 humid air (red crosses) accompanied by the blue dashed line separating the PICC and 5 RICC components of ΔK, as defined in Section 3.3. (b) – (d) light microscopy images of 6 the fracture surfaces of the corresponding specimens. The investigated area of crack 7 propagation is marked by the white vertical arrow and the crack arrest at the measured 8 threshold is underlined by the white dotted line. 9 10 1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1 2 4 8 16 32 da/dN[mm/cycle] ΔK[MPa·m0.5] R=0.8 humid air R=0.1 humid air R=0.1 dry air R=0.1 ΔKeff + PICC R = 0.1 dry air R = 0.1 humid air R = 0.8 humid air 1 mm (d) 1 mm (c) 1 mm (b) EA4T R = 0.8 humid air R = 0.1 humid air (a) R = 0.1 dry air R = 0.1 ΔKeff + ΔKPICC 23 1 2 Fig. 9. (a) Experimental crack growth rates and their fitting lines for the P265GH steel obtained 3 at R = 0.8 in humid air (green diamonds), at R = 0.1 in dry air (blue circles) and at R = 0.1 4 in humid air (red crosses) accompanied by the blue dashed line separating the PICC and 5 RICC components of ΔK, as defined in Section 3.3. (b) – (d) light microscopy images of 6 the fracture surfaces of the corresponding specimens. The investigated area of crack 7 propagation is marked by the white vertical arrow and the crack arrest at the measured 8 threshold is underlined by the white dotted line. 9 10 11 12 1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1 2 4 8 16 32 da/dN[mm/cycle] ΔK[MPa·m0.5] R=0.8 humid air R=0.1 humid air R=0.1 dry air R=0.1 ΔKeff + PICC R = 0.1 dry air R = 0.1 humid air R = 0.8 humid air 1 mm (d) 1 mm (b) 1 mm (c) P265GH R = 0.8 humid air R = 0.1 humid air (a) R = 0.1 dry air R = 0.1 ΔKeff + ΔKPICC 24 3.4 Extraction of important parameters and decomposition of ΔKth 1 As described in Sections 2.2, 2.3 and 2.4, the parameters of the fitting Eqs. (2) and (3) provide useful 2 information about the composition of ΔK in relation to driving force and closure. Table 3 presents the 3 obtained fitting parameters. The values then enabled determination of the crack closure parameter f 4 according to Eq. (6) and the crack closure components for threshold loading ΔKPICC,th, ΔKRICC,th and 5 ΔKOICC,th according to Eqs. (8), (10) and (14), respectively. The results are summarized in Table 4. 6 The components ΔKeff,th, ΔKPICC,th, ΔKRICC,th and ΔKOICC,th were graphically expressed in Fig. 10 to 7 offer a much better overview of their significance and the corresponding resistance to fatigue crack 8 growth. Owing to this diagram, the responsible mechanisms forming the total threshold can be clearly 9 imagined. Although slight shifts of some of the components can be subject to discussion, such 10 decomposition is unprecedented and offers a unique insight into the complex problem of fatigue crack 11 growth threshold. The OICC component enables assessment of the range, in which the traditionally 12 defined threshold may vary due to experimental conditions. 13 Table 3: Fitting parameters of the crack growth rate Eqs. (2) and (3) obtained based on the 14 experimental data. 15 Material Ceff [*] CR=0.1 [*] n [*] p [–] ΔKeff,th [MPa·m0.5] ΔKR=0.1,th (dry air) [MPa·m0.5] ΔKR=0.1,th (humid air) [MPa·m0.5] X20Cr13 7.0·10–9 3.3·10–9 3.2 0.65 2.8 4.8 5.7 GX4CrNi13-4 3.2·10–8 1.8·10–8 2.4 0.75 2.7 3.6 4.5 EA1N 1.7·10–8 1.3·10–8 2.8 0.60 3.0 4.6 5.2 EA4T 2.0·10–8 1.5·10–8 2.6 0.70 2.5 4.2 6.1 P265GH 1.3·10–8 1.0·10–8 2.8 0.70 3.0 5.0 7.0 *Units of C and n correspond to da/dN values in [mm/cycle]. 16 17 Table 4: Crack closure parameters calculated using Eqs. (6), (8), (11) and (14). 18 Material f = KPICC/Kmax ΔKPICC,th [MPa·m0.5] ΔKRICC,th [MPa·m0.5] ΔKOICC,th [MPa·m0.5] X20Cr13 0.29 1.01 1.04 0.9 GX4CrNi13-4 0.29 0.77 0.13 0.9 EA1N 0.18 0.42 1.18 0.6 EA4T 0.19 0.44 1.26 1.9 P265GH 0.18 0.45 1.55 2.0 19 20 21 22 23 24 25 1 2 Fig. 10. Graphical summary of the threshold ΔKth decomposition with respect to the mechanisms 3 for the five investigated materials. The effective threshold ΔKeff,th, the plasticity-induced 4 crack closure component ΔKPICC,th, the roughness-induced crack closure component 5 ΔKRICC,th and the oxide-induced crack closure component ΔKOICC,th. 6 7 4 Discussion 8 4.1 Significance of the results and general discussion 9 Experimental techniques of crack closure measurement, such as the back face or crack mouth opening 10 displacement techniques are not reliable and many discussions and questions arise about the relevance of 11 the results. While they still represent a way to obtain the values, especially under variable-amplitude 12 loading, the presented method offers to find the crack closure parameters without any problematic 13 measurement techniques. Moreover, the possibility of separation to individual closure components is 14 offered. 15 4.1.1 PICC evaluated from crack growth rates versus numerical modelling 16 One of the interesting findings of the presented procedure is that the characteristic crack closure 17 parameter f for the 5 investigated materials were obtained directly from the crack growth rates without 18 any necessity of measurement or modelling of crack closure. These values are not affected by any 19 problematic aspects of the crack closure theory in the sense of the direct contact determination 20 techniques. Looking at the results in Table 4, two groups of materials with similar numbers were formed, 21 one for the non-corroding steels (f ≈ 0.3) and one for the corroding steels (f ≈ 0.2). The reason can only be 22 a subject of speculation. It is clear, though, that without the newly proposed methodology, this 23 information about the materials would be lost and the true f values could not be compared in this way. 24 They would be computed as ≈ 0.25 for all materials. The presented methodology not only safes time and 25 effort by avoiding crack closure measurement but it also removes the need for finite element modelling of 26 PICC. These numerical models are typically too complex with high demands on computational time with 27 insufficient ability to predict. Many parameters of the simulation need to be calibrated, in which case the 28 experimental data for the material need to be available anyway. The reliability of crack closure 29 experimental measurement has been subject to debate for a long time with no consensus. The present 30 0 1 2 3 4 5 6 7 8 Composition of ΔKth [MPa·m0.5] ΔK_OICC,th ΔK_RICC,th ΔK_PICC,th ΔK_eff,th (R = 0.1) ΔKOICC,th ΔKRICC,th ΔKPICC,th ΔKeff,th 32 [11] Kujawski D, Vasudevan AK. 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